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受欢迎的三角函数 >

证明 tan(θ)cot(θ)-1=1-sec(θ)cos(θ)

解答

证明

tan (θ) cot (θ) - 1 = 1 - sec (θ) cos (θ)

解答

真

求解步骤

tan (θ) cot (θ) - 1 = 1 - sec (θ) cos (θ)

调整左侧

tan (θ) cot (θ) - 1

用 sin, cos 表示

- 1 + cot (θ) tan (θ)

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 1 + \frac{cos ( θ )}{sin ( θ )} tan (θ)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + \frac{cos ( θ )}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

- 1 + \frac{cos ( θ )}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )} = 0

- 1 + \frac{cos ( θ )}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )} = 1

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{cos ( θ ) sin ( θ )}{sin ( θ ) cos ( θ )}

约分:

cos (θ)

= \frac{sin ( θ )}{sin ( θ )}

约分:

sin (θ)

= 1

= - 1 + 1

化简

= 0

= 0

= 0

调整右侧

1 - sec (θ) cos (θ)

用 sin, cos 表示

1 - cos (θ) sec (θ)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

= 1 - cos (θ) \frac{1}{cos ( θ )}

1 - cos (θ) \frac{1}{cos ( θ )} = 0

1 - cos (θ) \frac{1}{cos ( θ )}

cos (θ) \frac{1}{cos ( θ )} = 1

cos (θ) \frac{1}{cos ( θ )}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( θ )}{cos ( θ )}

约分:

cos (θ)

= 1

= 1 - 1

化简

= 0

= 0

= 0

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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